Simulations As Crucial Tools for GW Experiments

LIGO-P040046-00-E

Simulations as crucial tools for GW experiments

Hiroaki Yamamoto
LIGO Laboratory, California Institute of Technology, Pasadena, CA 91125

In order to operate large scale GW detection experiments, like the LIGO project, use of computer simulations of various kinds is crucial. In this presentation, two kinds of simulation tools used in the LIGO project are explained. One is a time domain simulation, which generates time series of various channels and is used to study nonstationary processes. Another is a static optics model, which calculates steady state fields in a given optical configuration and is used to study details of stationary processes.

I. INTRODUCTION

EVERAL large scale experiments to directly detect gravitational wave signal using the laser interferometry technology have started operating or are in important commissioning phases. These experimental systems are consisted of various subsystems each of which is highly demanding and tight integrations among them are very important [1],[2].

In order to design such a complex integrated system and to operation commission its operation at its the design performance, various simulation tools have been developed. One is a time domain simulation [3].[4], which tries to trace traces the time evolution of the system as realistic realistically as possible. The main use of this software is a the study of time dependent processes, like the lock acquisition process or control system designs where gain constants changes may change dynamically due to nonstationarity of the system. Because the time domain simulation traces each step of the changes of the system, the program usually runs slowtypically is very CPU-intensive.

In order to make the time domain simulation speed acceptable, various simplifications are done introduced by compromisingtrading off computational efficiency the against accuracy. Static simulation programs complement this shortcoming of the time domain simulation [5],[6]. The static program calculates the steady state fields in a specified optical configuration. This kind of program can incorporates incorporate many details of the optical systems, like the aberration of the mirror surface. The static simulation is usually used to understand the details of the as-built optical systems, like the degradation of the performance due to surface aberration and mirror deformation due to thermal heating.

In order to express the spatial beam profile, there are two major different waysexpansions that are used. One is to expand using sinusoidal modes (e.g., the Fast Fourier Transform) [5][6] and the other one is to expand using Hermite- Gaussian functions, called Modal Model [7]. Most of the time domain simulation programs uses use a finite number of Hermite Gaussian modes to trace the time evolution of the beam profile, while both methods are used for the static field calculation [5][6][8].

In this paper, the time domain simulation used in LIGO, called the End- to- End simulation framework [4], is explaineddescribed. Following that section, a static field simulation using the sinusoidal expansion [6] is explaineddescribed. The application of these simulation softwares for theto LIGO commissioning and to the future interferometer design are discussed.

II. LIGO End to End Simulation

A. Overview

The time domain simulation program used in LIGO, “LIGO End to End model”, hereafter called e2e, was developed at by LIGO Lab Laboratory in Caltech for the design of the LIGO detector interferometers and to assist it’s in their commissioning.

e2e is written in C++, and it provides an environment and tools for simulating LIGO LIGO-like interferometer systems. The software is designed to model wide varieties of systems, from a simple pendulum to a complete interferometer with complex opto-mechanical systems connected by delicate control systems. It is intended to be used for wide varieties of purposes, from a prototyping of a subsystem to a full scale simulation of a performance of an entire interferometer.

The C++ class structure used in e2e is based on an object oriented design, and it allows the implementation of new functionalities independent from the rest of the code and the handling of the time evolution is transparent. This design makes it possible that for a scientist who wants to add a new physics can to concentrate on the implementation of the specific topics, instead of worrying how to make the new code working work properly in a complex time domain simulation.

The e2e software package is similar to MATLAB [9], i.e., it provides an environment in which you can design a system to simulate by combining built-in modules. Most of the important tools to construct a LIGO-like experiment are provided as built-in tools, which are discussed in the next section. When necessary, a new functionality can be easily added by providing C++ codes. A graphical front end is also provided.

B. Building blocks

The e2e framework has modules to build interferometer systems. All modules are explained in the e2e reference manual [10]. Several A number of examples of the built-in modules are given in the following table.

Table I

Examples of build-in modules

name / functionality
field_gen / Laser source. Amplitude and frequency noises can be added, and the spatial beam profile can be specified by Hermite Gaussian modes.
mirror2 / Mirror with two input fields and two output fields. The displacement and orientation can be specified.
pd_demod / Photo diode and RF mixer, with low pass filter. An arbitrary detector shape can be specified and the shot noise can be simulated.
rec_sum / Fast simulation of a power recycled Michelson cavity.
susp3DMass / Suspended mass with a single pendulum, with the suspension point and actuation force as inputs.
FUNC_X / An interface to easily embed C++ code in the simulation.

By combining these modules, one can define a system to simulate. Once a system is defined, it can be saved as a module, and it can then be used in integrated into other modules just as built-in modules.

An The analog continuous time evolution is simulated by a digitized discrete time evolution with a very small time step, which is usually a one-way light transit trip time of a laser in for the long arm, which. This corresponds is to 13ms for LIGO. Linear responses are simulated using a digital filter or a state space model. Digital control systems are simulated by properly digitizing the time response. Because the analog continuous process is approximated by a small time step digital discrete process, the simulation time step needs to be chose chosen to be smaller small enough enough compared with to the band widthbandwidth of the simulating system.

Any optical systems with optics placed on a planearranged in a plane can be simulated by combining mirror and field propagation modules. In order to simulate a resonating resonant cavity, the time step needs to be chosen to be the one-way trip light transit time of the cavity. For an interferometer using a power recycled Michelson cavity (PRM), lengths of the PRM cavity determines the time step, which makes the simulation very slow. In order to work around this constraint, the response of the PRM is calculated using an adiabatic approximation that where all fields and mirrors positions do not change while a laser moves one way through the long armduring a time interval corresponding to the cavity light transit time [11]. With this approximation, the simulation time step is not dictated determined by the short cavity of the PRM.

There are functionalities models which are impossible or inconvenient to build using existing modules. E.g., a lock acquisition process needs a complicated algorithm. A Therefore, a module, FUNC_X, is provided to easily integrate new functionality to into the e2e environment [12]. With the FUNC_X module, one can embed any C++ code as a part of the simulation. Interactions with other modules and book keeping as a part of the time domain simulation loop are automaticallytransparently taken carehandled by the e2e framework, and the code can be developed only by concentrating on the physics of interest.

C. Interface

In order to prepare a file to define the system to be simulated, a graphical user interface is provided, analogous to the Simulink package of MATLAB. By using an iconic interface, one can intuitively design and maintain a system to be simulated. Various settings of each module can be defined using this interface as well. Names of settings, default values and short descriptions are displayed in the setting dialog, and the manual of modules are rarely.

III. Application of time domain simualtion

A. Overview

The major application of the e2e simulation was the design of the code to acquire lock of LIGO [13][14]. The full simulation of the LIGO system was developed, which can properly generate the sensitivity curve of the locked state LIGO.

In the LIGO project, e2e was not use used in real time or online connected with the actual hardware environment. The flexibility of the e2e environment was fully utilized to test various possibilities, like different design of control systems, in a reasonably realistic condition, i.e., noisy and non stationary environment. After a guideline was understood, the actual development was done at the site utilizing the knowledge learned from the simulation.

In addition to the two applications detailed below, it served to identify various issues. The misplacement incorrect value of the Shnupp Schnupp asymmetry of the 4km LIGO interferometers was pointed discovered out through simulation during the a study of the locking at higher power operation. The miscalculation of the wave front sensor matrix was pointed outalso discovered through simulation. The problem was identified because the e2e simulation includes automatically includes all necessary ingredients physics with reasonable accuracy.

B. Lock acquisition

The first important application of e2e was the design of the lock acquisition code of the 2km LIGO detector interferometer at Hanford. A custom software package simulation package for this the 2km interferometer was developed using within the e2e framework. This system, called Han2k, is based on a ratherimplements a simplified, but model, yet containing all the crucial physics [15]:.

· Only the length degree of freedom was taken into account.

· Essential structure of the length control system is was simulated without implementing digital control systems.

· Laser fields were approximated by as plane waves.

· Strong Common-mode seismic correlations in at the low frequency regionfrequencies were assumed among seismic for motions of the four stacks forming the Michelson chamberinterferometer.

· Each mirror is was approximated by a simple point-like object suspended by a single wire, moving only along the direction of the beam.

· Saturation of actuation forces on mirrors is was included.

Using this model, the lock acquisition code was developed. The cC code developed derived in from the simulation framework was then used in the real time control systems of the interferometer at the site to lock the interferometer. After the interferometer was successfully locked, it was confirmed that the simulation could reproduce characteristics of various error signals and power build ups up during the lock acquisition process [13].

C. In-lock state sensitivity

After a fewDuring the years of commissioning since the first lock of LIGO, the LIGO interferometer sensitivity has come very close to the design performance [16]. In order to understand the details of the LIGO performance and to help to improve designs of various control systems, a more sophisticated LIGO simulation program, called SimLIGO, has been developed [17].

The major characteristics of this model are as follows:

· Length and alignment degree of freedoms are taken into account.

· Essential lengths and alignment control systems are simulated using digital control systems.

· Laser fields are approximated using a time domain modal model [18], up to 4th order modes, to simulate the spatial beam profile.

· Frequency dependent correlations among stacks are modeled.

· Each mirror is was modeled by a 3 dimensional palate cylinder suspended by a stiff single wire. Bounce modes are included in an ad hoc way to match with measurements.

· The sensing and actuation of the mirror are simulated as close realistically as possible to the reality, including allowing for a possible imbalance of the 4 sensors and actuators on one mirror.

Figure 1 shows the comparison of the design sensitivity, the measured LIGO sensitivity and the e2e simulation prediction. Overall, the simulation can reproduce the basic characteristics of the sensitivity.

Fig. 1. The LIGO Interferometer sensitivity. The thick line is the design sensitivity, the thin line is the sensitivity measured on August 15th, 2004, and the dashed line the SimLIGO simulation prediction.

The differenceDifferences at the low frequency region end of the spectrum will will comearise from the inaccurate parameterization of the seismic isolation system in e2e, which was identified recently. The difference in at the high frequency end will comearise from the fact that mirrors used in the e2e simulation are assumed to be perfect, without aberration. The static model simulation discussed next showed descentis able to account for shot-noise degradation degradation due when as-built to the mirror surface surface figure aberrationaberrations are included.

D. For theApplications to future interferometerinterferometers

Several projects, including LIGO, have started designing the next generation interferometer [2],[19]. The simulation tools like such as e2e is are very important for the design of those future interferometers.

The e2e framework has all the tools needed to simulate the advanced interferometers. A few works efforts have already started [20]. ButHowever, there are several a number of improvements and additions extensions to the model that are needed to make e2e practically usableuseful. Most The most important addition is the fast simulation of the dual recycling Michelson cavity. The formulation has been done completed [21] and the module is now under development.